How to stick a curve (with shape keys) on rig?
You can't assign curves to armatures directly nor can you weight paint.

Only workaround I know is using Hooks

You can either select your armature then enter Edit Mode on your curve, select all desired vertex and assign a hook, or manually add a hook modifier and assing the vertex afterwards.

You can hook to the whole armature, or you can hook to individual bones. Matching rotations will be an issue, but it should keep any shape keys working.